24a_blackscholes

24a_blackscholes - Econ 251a Fall 2006 John Geanakoplos...

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Econ 251a Fall 2006 John Geanakoplos Dynamically Hedging Call Options Abstract: Hedging and Pricing are simultaneous operations. 1 Pricing Consider a stock selling for $100. Over the course of a year it is expected to increase in value by 10%, with a standard deviation of 16%. These are typical numbers for reason the stock price is only $100, instead of $110/1.06, is the risk. In the states next year when the stock price is well above $110, the economy as a whole is probably also doing well, so the marginal investor is eating pretty well, and so his marginal utility for money is relatively low. In the states when the stock price next year is well below $110, the economy is probably doing relatively badly, and the typical consumer has a high marginal utility of money. Thus the stock pays most when the money is needed least, and so it is worth less than an asset which delivered $110 in every state. Consider a call option on this stock with expiration date one year into the future, with exercise price 106. (Note that the present value of the exercise price is equal to the stock price, which is the case for which the call option has the same value as the put option, and also is a special case where the Black-Scholes formula for the call option price is particularly simple). It can be shown that the expected payo f from the call option is about $9.4(1.06). One might think therefore that the call option should sell for $9.4. But that is wrong. The call option pays most when the stock price is highest, which we have seen is when the marginal utility of money is lowest. Hence the call option should sell for less than $9.4. But for how much less? One does not seem to have enough information to answer this question. Suppose that (1) the interest rate will never change, and that (2) the stock price moves continuously through time, and that (3) the percentage changes in the stock price are identically distributed over equally long intervals, and that (4) the percent- age change in stock price over any time interval is independent of the percentage change in stock price over any other disjoint time interval. Condition (2) means that there are no jumps in the stock price, such as there would be in a Poisson process. Condition (3) means that the probability of a 1% (or 12%) increase in the stock price over one day is the same as the probabiity of a 1% (or 12%) increase over any other day, and that the probability of a 1% (or 12%) increase over one hour is the same as the probability of a 1% (or 12%) increase over any other other hour. Condition (4) says that knowing the percentage change in any one day is of no help in predicting the 1
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percentage change over any other day. One describes these conditions by saying that the stock follows a geometric Brownian motion, or that the log of the price follows a random walk. It can be shown that if the market is con
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24a_blackscholes - Econ 251a Fall 2006 John Geanakoplos...

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